:: CLVECT_1 semantic presentation

begin

definition
attr c1 is strict ;
struct CLSStruct -> ( ( ) ( ) addLoopStr ) ;
aggr CLSStruct(# carrier, ZeroF, addF, Mult #) -> ( ( strict ) ( strict ) CLSStruct ) ;
sel Mult c1 -> ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of c1 : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ;
end;

registration
cluster non empty for ( ( ) ( ) CLSStruct ) ;
end;

definition
let V be ( ( ) ( ) CLSStruct ) ;
mode VECTOR of V is ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
end;

definition
let V be ( ( non empty ) ( non empty ) CLSStruct ) ;
let v be ( ( ) ( ) VECTOR of V : ( ( non empty ) ( non empty ) CLSStruct ) ) ;
let z be ( ( complex ) ( complex ) Complex) ;
func z * v -> ( ( ) ( ) Element of ( ( ) ( ) set ) ) equals :: CLVECT_1:def 1
 the Mult of V : ( ( ) ( ) NORMSTR ) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) . [z : ( ( Function-like V18([:V : ( ( ) ( ) NORMSTR ) ,V : ( ( ) ( ) NORMSTR ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like [:V : ( ( ) ( ) NORMSTR ) ,V : ( ( ) ( ) NORMSTR ) :] : ( ( ) ( ) set ) -defined V : ( ( ) ( ) NORMSTR ) -valued Function-like V18([:V : ( ( ) ( ) NORMSTR ) ,V : ( ( ) ( ) NORMSTR ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) NORMSTR ) ) ) Element of bool [:[:V : ( ( ) ( ) NORMSTR ) ,V : ( ( ) ( ) NORMSTR ) :] : ( ( ) ( ) set ) ,V : ( ( ) ( ) NORMSTR ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of V : ( ( ) ( ) NORMSTR ) ) ] : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;
end;

registration
let ZS be ( ( non empty ) ( non empty ) set ) ;
let O be ( ( ) ( ) Element of ZS : ( ( non empty ) ( non empty ) set ) ) ;
let F be ( ( Function-like V18([:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined ZS : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) BinOp of ZS : ( ( non empty ) ( non empty ) set ) ) ;
let G be ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined ZS : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ;
cluster CLSStruct(# ZS : ( ( non empty ) ( non empty ) set ) ,O : ( ( ) ( ) Element of ZS : ( ( non empty ) ( non empty ) set ) ) ,F : ( ( Function-like V18([:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined ZS : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) Element of bool [:[:ZS : ( ( non empty ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,G : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined ZS : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) ) ) Element of bool [:[:COMPLEX : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,ZS : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) #) : ( ( strict ) ( strict ) CLSStruct ) -> non empty strict ;
end;

definition
let IT be ( ( non empty ) ( non empty ) CLSStruct ) ;
attr IT is vector-distributive means :: CLVECT_1:def 2
 for a being ( ( complex ) ( complex ) Complex)
for v, w being ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) set ) ) holds a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) * (v : ( ( complex ) ( complex ) Complex) + w : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) ) : ( ( ) ( ) Element of the carrier of IT : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = (a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) * v : ( ( complex ) ( complex ) Complex) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) + (a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) * w : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of IT : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;
attr IT is scalar-distributive means :: CLVECT_1:def 3
 for a, b being ( ( complex ) ( complex ) Complex)
for v being ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) set ) ) holds (a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) + b : ( ( complex ) ( complex ) Complex) ) : ( ( ) ( ) set ) * v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = (a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) * v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) + (b : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of IT : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ;
attr IT is scalar-associative means :: CLVECT_1:def 4
 for a, b being ( ( complex ) ( complex ) Complex)
for v being ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) set ) ) holds (a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) * b : ( ( complex ) ( complex ) Complex) ) : ( ( ) ( ) set ) * v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = a : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) * (b : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ;
attr IT is scalar-unital means :: CLVECT_1:def 5
 for v being ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) set ) ) holds 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) * v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = v : ( ( ) ( ) VECTOR of IT : ( ( non empty ) ( non empty ) CLSStruct ) ) ;
end;

definition
func Trivial-CLSStruct -> ( ( strict ) ( strict ) CLSStruct ) equals :: CLVECT_1:def 6
 CLSStruct(# 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ,op0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal ) Element of 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) ,op2 : ( ( Function-like V18([:1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) ) ( Relation-like [:1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty ) set ) -defined 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -valued Function-like V18([:1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) ) Element of bool [:[:1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(pr2 (COMPLEX : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) )) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty V50() ) set ) -defined 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) ) Element of bool [:[:COMPLEX : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty V50() ) set ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) #) : ( ( strict ) ( non empty strict ) CLSStruct ) ;
end;

registration
cluster Trivial-CLSStruct : ( ( strict ) ( strict ) CLSStruct ) -> 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element strict ;
end;

registration
cluster non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for ( ( ) ( ) CLSStruct ) ;
end;

definition
mode ComplexLinearSpace is ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ;
end;

theorem :: CLVECT_1:1
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) st ( z : ( ( complex ) ( complex ) Complex) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) or v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) holds
z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:2
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds
( not z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) or z : ( ( complex ) ( complex ) Complex) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) or v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: CLVECT_1:3
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds - v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:4
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = - v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:5
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:6
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds z : ( ( complex ) ( complex ) Complex) * (- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = (- z : ( ( complex ) ( complex ) Complex) ) : ( ( complex ) ( complex ) set ) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:7
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds z : ( ( complex ) ( complex ) Complex) * (- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = - (z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:8
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds (- z : ( ( complex ) ( complex ) Complex) ) : ( ( complex ) ( complex ) set ) * (- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:9
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds z : ( ( complex ) ( complex ) Complex) * (v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = (z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - (z : ( ( complex ) ( complex ) Complex) * u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:10
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z1, z2 being ( ( complex ) ( complex ) Complex) holds (z1 : ( ( complex ) ( complex ) Complex) - z2 : ( ( complex ) ( complex ) Complex) ) : ( ( ) ( complex ) set ) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = (z1 : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) - (z2 : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:11
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) st z : ( ( complex ) ( complex ) Complex) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) & z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = z : ( ( complex ) ( complex ) Complex) * u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:12
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z1, z2 being ( ( complex ) ( complex ) Complex) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) <> 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) & z1 : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = z2 : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) holds
z1 : ( ( complex ) ( complex ) Complex) = z2 : ( ( complex ) ( complex ) Complex) ;

theorem :: CLVECT_1:13
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for z being ( ( complex ) ( complex ) Complex)
for F, G being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) st len F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) = len G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) & ( for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) )
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) in dom F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( non empty V50() ) set ) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( ) set ) holds
F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( ) set ) = z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) holds
Sum F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = z : ( ( complex ) ( complex ) Complex) * (Sum G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V50() FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:14
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for z being ( ( complex ) ( complex ) Complex) holds z : ( ( complex ) ( complex ) Complex) * (Sum (<*> the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V50() cardinal {} : ( ( ) ( ) set ) -element FinSequence-like FinSubsequence-like FinSequence-membered ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:15
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds z : ( ( complex ) ( complex ) Complex) * (Sum <*v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ,u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like non empty V50() 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = (z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + (z : ( ( complex ) ( complex ) Complex) * u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:16
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v1, v2 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex) holds z : ( ( complex ) ( complex ) Complex) * (Sum <*u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ,v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ,v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like non empty V50() 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = ((z : ( ( complex ) ( complex ) Complex) * u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + (z : ( ( complex ) ( complex ) Complex) * v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + (z : ( ( complex ) ( complex ) Complex) * v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:17
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds Sum <*v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ,v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like non empty V50() 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = (1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) + 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:18
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds Sum <*(- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ,(- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like non empty V50() 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = (- (1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) + 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:19
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds Sum <*v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ,v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ,v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like non empty V50() 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element FinSequence-like FinSubsequence-like ) FinSequence of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = ((1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) + 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) + 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

begin

definition
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
let V1 be ( ( ) ( ) Subset of ) ;
attr V1 is linearly-closed means :: CLVECT_1:def 7
( ( for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty ) ( non empty ) set ) ) st v : ( ( complex ) ( complex ) Complex) in V1 : ( ( ) ( ) set ) & u : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V1 : ( ( ) ( ) set ) holds
v : ( ( complex ) ( complex ) Complex) + u : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) in V1 : ( ( ) ( ) set ) ) & ( for z being ( ( complex ) ( complex ) Complex)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty ) ( non empty ) set ) ) st v : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V1 : ( ( ) ( ) set ) holds
z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) in V1 : ( ( ) ( ) set ) ) );
end;

theorem :: CLVECT_1:20
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( ) set ) & V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) ;

theorem :: CLVECT_1:21
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V1 : ( ( ) ( ) Subset of ) holds
- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) ;

theorem :: CLVECT_1:22
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V1 : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V1 : ( ( ) ( ) Subset of ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in V1 : ( ( ) ( ) Subset of ) ;

theorem :: CLVECT_1:23
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) holds {(0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty trivial 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element ) Element of bool the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-closed ;

theorem :: CLVECT_1:24
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) = V1 : ( ( ) ( ) Subset of ) holds
V1 : ( ( ) ( ) Subset of ) is linearly-closed ;

theorem :: CLVECT_1:25
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1, V2, V3 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed & V2 : ( ( ) ( ) Subset of ) is linearly-closed & V3 : ( ( ) ( ) Subset of ) = { (v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) where v, u is ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V1 : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V2 : ( ( ) ( ) Subset of ) ) } holds
V3 : ( ( ) ( ) Subset of ) is linearly-closed ;

theorem :: CLVECT_1:26
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1, V2 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is linearly-closed & V2 : ( ( ) ( ) Subset of ) is linearly-closed holds
V1 : ( ( ) ( ) Subset of ) /\ V2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is linearly-closed ;

definition
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
mode Subspace of V -> ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) means :: CLVECT_1:def 8
( the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) c= the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) & 0. it : ( ( ) ( ) set ) : ( ( ) ( ) Element of the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = 0. V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) & the addF of it : ( ( ) ( ) set ) : ( ( Function-like V18([: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = the addF of V : ( ( non empty ) ( non empty ) set ) : ( ( Function-like V18([: the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) || the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) & the Mult of it : ( ( ) ( ) set ) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = the Mult of V : ( ( non empty ) ( non empty ) set ) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) | [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like ) Element of bool [:[:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of V : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: CLVECT_1:27
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
x : ( ( ) ( ) set ) in W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:28
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
x : ( ( ) ( ) set ) in V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;

theorem :: CLVECT_1:29
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for w being ( ( ) ( right_complementable ) VECTOR of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds w : ( ( ) ( right_complementable ) VECTOR of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) is ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:30
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds 0. W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:31
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds 0. W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) = 0. W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:32
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for w1, w2 being ( ( ) ( right_complementable ) VECTOR of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st w1 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & w2 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
w1 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) + w2 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:33
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for w being ( ( ) ( right_complementable ) VECTOR of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st w : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
z : ( ( complex ) ( complex ) Complex) * w : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) = z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:34
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for w being ( ( ) ( right_complementable ) VECTOR of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st w : ( ( ) ( right_complementable ) VECTOR of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = - w : ( ( ) ( right_complementable ) VECTOR of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:35
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for w1, w2 being ( ( ) ( right_complementable ) VECTOR of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st w1 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & w2 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
w1 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) - w2 : ( ( ) ( right_complementable ) VECTOR of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:36
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:37
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds 0. W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) in W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:38
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds 0. W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) in V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;

theorem :: CLVECT_1:39
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:40
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:41
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:42
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:43
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of )
for D being ( ( non empty ) ( non empty ) set )
for d1 being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) )
for A being ( ( Function-like V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) st V1 : ( ( ) ( ) Subset of ) = D : ( ( non empty ) ( non empty ) set ) & d1 : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) = 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) & A : ( ( Function-like V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b3 : ( ( non empty ) ( non empty ) set ) ) = the addF of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || V1 : ( ( ) ( ) Subset of ) : ( ( ) ( Relation-like Function-like ) set ) & M : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) = the Mult of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) | [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,V1 : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of bool [:[:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) holds
CLSStruct(# D : ( ( non empty ) ( non empty ) set ) ,d1 : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) ,A : ( ( Function-like V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b3 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) CLSStruct ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:44
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) holds V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:45
for V, X being ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & X : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) = X : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;

theorem :: CLVECT_1:46
for V, X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) st V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:47
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st the carrier of W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) c= the carrier of W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) holds
W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:48
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st ( for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

registration
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
cluster non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital for ( ( ) ( ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ;
end;

theorem :: CLVECT_1:49
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st the carrier of W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) = the carrier of W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) holds
W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:50
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st ( for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ) holds
W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:51
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st the carrier of W : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) = the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) holds
W : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;

theorem :: CLVECT_1:52
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st ( for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
W : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;

theorem :: CLVECT_1:53
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) = V1 : ( ( ) ( ) Subset of ) holds
V1 : ( ( ) ( ) Subset of ) is linearly-closed ;

theorem :: CLVECT_1:54
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) <> {} : ( ( ) ( ) set ) & V1 : ( ( ) ( ) Subset of ) is linearly-closed holds
ex W being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st V1 : ( ( ) ( ) Subset of ) = the carrier of W : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ;

definition
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
func (0). V -> ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) means :: CLVECT_1:def 9
 the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) = {(0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) : ( ( ) ( zero right_complementable ) Element of the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty trivial 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element ) Element of bool the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
func (Omega). V -> ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) equals :: CLVECT_1:def 10
 CLSStruct(# the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) , the ZeroF of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) , the addF of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( Function-like V18([: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the Mult of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) CLSStruct ) ;
end;

theorem :: CLVECT_1:55
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds (0). W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = (0). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:56
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds (0). W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = (0). W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:57
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds (0). W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:58
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds (0). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:59
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds (0). W1 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of W2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:60
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) holds V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) is ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of (Omega). V : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

definition
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
let v be ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;
func v + W -> ( ( ) ( ) Subset of ) equals :: CLVECT_1:def 11
 { (v : ( ( ) ( ) set ) + u : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) where u is ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) : u : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( Function-like V18([:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) ( Relation-like [:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) -defined V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) -valued Function-like V18([:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) Element of bool [:[:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) } ;
end;

definition
let V be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
let W be ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;
mode Coset of W -> ( ( ) ( ) Subset of ) means :: CLVECT_1:def 12
 ex v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) st it : ( ( Function-like V18([:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) ( Relation-like [:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) -defined V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) -valued Function-like V18([:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) Element of bool [:[:V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( ) set ) : ( ( ) ( ) Subset of ) ;
end;

theorem :: CLVECT_1:61
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:62
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ;

theorem :: CLVECT_1:63
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds (0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ;

theorem :: CLVECT_1:64
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + ((0). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = {v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) } : ( ( ) ( non empty trivial 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element ) Element of bool the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:65
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + ((Omega). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ;

theorem :: CLVECT_1:66
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:67
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:68
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
(z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ;

theorem :: CLVECT_1:69
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st z : ( ( complex ) ( complex ) Complex) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) & (z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:70
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) iff (- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:71
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = (v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: CLVECT_1:72
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = (v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: CLVECT_1:73
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: CLVECT_1:74
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = (- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:75
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v1, v2 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) holds
v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ;

theorem :: CLVECT_1:76
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in (- v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:77
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st z : ( ( complex ) ( complex ) Complex) <> 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V50() ) set ) ) & z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) holds
v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:78
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for z being ( ( complex ) ( complex ) Complex)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ;

theorem :: CLVECT_1:79
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( - v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:80
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:81
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:82
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff ex v1 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: CLVECT_1:83
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff ex v1 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: CLVECT_1:84
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v1, v2 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( ex v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) & v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ) iff v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:85
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) holds
ex v1 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:86
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) holds
ex v1 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:87
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W1, W2 being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) iff W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:88
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v, u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W1, W2 being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) holds
W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:89
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
( C : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) is linearly-closed iff C : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:90
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W1, W2 being ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for C1 being ( ( ) ( ) Coset of W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) )
for C2 being ( ( ) ( ) Coset of W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st C1 : ( ( ) ( ) Coset of b2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = C2 : ( ( ) ( ) Coset of b3 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
W1 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) = W2 : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ;

theorem :: CLVECT_1:91
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds {v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) } : ( ( ) ( non empty trivial 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element ) Element of bool the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Coset of (0). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:92
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is ( ( ) ( ) Coset of (0). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
ex v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st V1 : ( ( ) ( ) Subset of ) = {v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) } : ( ( ) ( non empty trivial 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) -element ) Element of bool the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:93
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) is ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:94
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) holds the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) is ( ( ) ( ) Coset of (Omega). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:95
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of ) st V1 : ( ( ) ( ) Subset of ) is ( ( ) ( ) Coset of (Omega). V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( strict ) ( non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
V1 : ( ( ) ( ) Subset of ) = the carrier of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ;

theorem :: CLVECT_1:96
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
( 0. V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in C : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) iff C : ( ( ) ( ) Coset of b2 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = the carrier of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:97
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
( u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) iff C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( ) Subset of ) ) ;

theorem :: CLVECT_1:98
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
ex v1 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) + v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:99
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u, v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) & v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
ex v1 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:100
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for v1, v2 being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
( ex C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st
( v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) & v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b4 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ) iff v1 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) - v2 : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) in W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

theorem :: CLVECT_1:101
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for u being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for W being ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for B, C being ( ( ) ( ) Coset of W : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) st u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in B : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) & u : ( ( ) ( right_complementable ) VECTOR of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) in C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) holds
B : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) = C : ( ( ) ( ) Coset of b3 : ( ( ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) Subspace of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) ) ;

begin

definition
attr c1 is strict ;
struct CNORMSTR -> ( ( ) ( ) CLSStruct ) , ( ( ) ( ) N-ZeroStr ) ;
aggr CNORMSTR(# carrier, ZeroF, addF, Mult, normF #) -> ( ( strict ) ( strict ) CNORMSTR ) ;
end;

registration
cluster non empty for ( ( ) ( ) CNORMSTR ) ;
end;

definition
let IT be ( ( non empty ) ( non empty ) CNORMSTR ) ;
attr IT is ComplexNormSpace-like means :: CLVECT_1:def 13
 for x, y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for z being ( ( complex ) ( complex ) Complex) holds
( ||.(z : ( ( complex ) ( complex ) Complex) * x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) = |.z : ( ( complex ) ( complex ) Complex) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) * ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) & ||.(x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) + y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of IT : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <= ||.x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) + ||.y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) );
end;

registration
cluster non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital strict ComplexNormSpace-like for ( ( ) ( ) CNORMSTR ) ;
end;

definition
mode ComplexNormSpace is ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) CNORMSTR ) ;
end;

theorem :: CLVECT_1:102
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) holds ||.(0. CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) ) : ( ( ) ( zero right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V50() cardinal {} : ( ( ) ( ) set ) -element FinSequence-membered ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ;

theorem :: CLVECT_1:103
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds ||.(- x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) = ||.x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:104
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <= ||.x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) + ||.y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:105
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) <= ||.x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:106
for z1, z2 being ( ( complex ) ( complex ) Complex)
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds ||.((z1 : ( ( complex ) ( complex ) Complex) * x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) + (z2 : ( ( complex ) ( complex ) Complex) * y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b3 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <= (|.z1 : ( ( complex ) ( complex ) Complex) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) * ||.x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) + (|.z2 : ( ( complex ) ( complex ) Complex) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) * ||.y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:107
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) iff x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: CLVECT_1:108
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) = ||.(y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:109
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds ||.x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) - ||.y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <= ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:110
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds abs (||.x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) - ||.y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <= ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:111
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, w, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - w : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <= ||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) + ||.(y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - w : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) ;

theorem :: CLVECT_1:112
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x, y being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) <> y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds
||.(x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - y : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ;

definition
let CNS be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ;
let S be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ;
let z be ( ( complex ) ( complex ) Complex) ;
func z * S -> ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) means :: CLVECT_1:def 14
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) holds it : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) -defined CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) Element of bool [:[:COMPLEX : ( ( ) ( non empty V50() ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) = z : ( ( Function-like V18([:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) ( Relation-like [:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) -defined CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) -valued Function-like V18([:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) Element of bool [:[:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) * (S : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;
end;

definition
let CNS be ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) ;
let S be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ;
attr S is convergent means :: CLVECT_1:def 15
 ex g being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) st
for r being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) < r : ( ( ) ( complex real ext-real ) Real) holds
ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) holds
||.((S : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) - g : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) < r : ( ( ) ( complex real ext-real ) Real) ;
end;

theorem :: CLVECT_1:113
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S1, S2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent & S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) + S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) is convergent ;

theorem :: CLVECT_1:114
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S1, S2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent & S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) is convergent ;

theorem :: CLVECT_1:115
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) )
for S being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) is convergent ;

theorem :: CLVECT_1:116
for z being ( ( complex ) ( complex ) Complex)
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
z : ( ( complex ) ( complex ) Complex) * S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent ;

theorem :: CLVECT_1:117
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
||.S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , REAL : ( ( ) ( non empty V50() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined REAL : ( ( ) ( non empty V50() ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , REAL : ( ( ) ( non empty V50() ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ,REAL : ( ( ) ( non empty V50() ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) is convergent ;

definition
let CNS be ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) ;
let S be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ;
assume S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent ;
func lim S -> ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) means :: CLVECT_1:def 16
 for r being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) < r : ( ( ) ( complex real ext-real ) Real) holds
ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) holds
||.((S : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) - it : ( ( Function-like V18([:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) ( Relation-like [:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) -defined CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) -valued Function-like V18([:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ) ) Element of bool [:[:CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) ,CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of CNS : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) CLSStruct ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) < r : ( ( ) ( complex real ext-real ) Real) ;
end;

theorem :: CLVECT_1:118
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for g being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) )
for S being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) = g : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.(S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - g : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) .|| : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , REAL : ( ( ) ( non empty V50() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined REAL : ( ( ) ( non empty V50() ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , REAL : ( ( ) ( non empty V50() ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ,REAL : ( ( ) ( non empty V50() ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) is convergent & lim ||.(S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - g : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) .|| : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , REAL : ( ( ) ( non empty V50() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined REAL : ( ( ) ( non empty V50() ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , REAL : ( ( ) ( non empty V50() ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ,REAL : ( ( ) ( non empty V50() ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V50() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real V50() cardinal ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) ) ) ;

theorem :: CLVECT_1:119
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S1, S2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent & S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
lim (S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) + S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) = (lim S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) + (lim S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:120
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S1, S2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent & S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
lim (S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) = (lim S1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - (lim S2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:121
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for x being ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) )
for S being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
lim (S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) = (lim S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:122
for z being ( ( complex ) ( complex ) Complex)
for CNS being ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace)
for S being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) st S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
lim (z : ( ( complex ) ( complex ) Complex) * S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) = z : ( ( complex ) ( complex ) Complex) * (lim S : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() cardinal limit_cardinal ) Element of bool REAL : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed discerning reflexive vector-distributive scalar-distributive scalar-associative scalar-unital ComplexNormSpace-like ) ComplexNormSpace) : ( ( ) ( non empty ) set ) ) ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( right_complementable ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: CLVECT_1:123
for z being ( ( complex ) ( complex ) Complex)
for D being ( ( non empty ) ( non empty ) set )
for d1 being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) )
for A being ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,D : ( ( non empty ) ( non empty ) set ) )
for V being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace)
for V1 being ( ( ) ( ) Subset of )
for v being ( ( ) ( right_complementable ) VECTOR of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) )
for w being ( ( ) ( ) VECTOR of CLSStruct(# D : ( ( non empty ) ( non empty ) set ) ,d1 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ,A : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) CLSStruct ) ) st V1 : ( ( ) ( ) Subset of ) = D : ( ( non empty ) ( non empty ) set ) & M : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = the Mult of V : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) ) | [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,V1 : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of bool [:[:COMPLEX : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) , the carrier of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) : ( ( ) ( non empty V50() ) set ) ) & w : ( ( ) ( ) VECTOR of CLSStruct(# b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ,b4 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) set ) ) ,b5 : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) CLSStruct ) ) = v : ( ( ) ( right_complementable ) VECTOR of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) holds
z : ( ( complex ) ( complex ) Complex) * w : ( ( ) ( ) VECTOR of CLSStruct(# b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ,b4 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) set ) ) ,b5 : ( ( Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of [:COMPLEX : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty V50() ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) CLSStruct ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = z : ( ( complex ) ( complex ) Complex) * v : ( ( ) ( right_complementable ) VECTOR of b6 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ComplexLinearSpace) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( non empty ) set ) ) ;